Theorem crnggrpd 20156

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20155	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20151	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2109  Grpcgrp 18865  CRingccrg 20143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-ring 20144  df-cring 20145

This theorem is referenced by:  fermltlchr  21439  psdmul  22053  psd1  22054  psdpw  22057  ply1chr  22193  ply1fermltlchr  22199  elrgspnsubrunlem1  33198  elrgspnsubrunlem2  33199  erlbr2d  33215  rlocaddval  33219  rloccring  33221  rloc0g  33222  rlocf1  33224  fracerl  33256  ressply1evls1  33534  evl1deg1  33545  evl1deg2  33546  evl1deg3  33547  ply1dg1rt  33548  vr1nz  33559  irngss  33682  irredminply  33706  algextdeglem4  33710  algextdeglem5  33711  aks6d1c1p3  42098  aks6d1c2lem4  42115  aks6d1c6lem2  42159  aks5lem2  42175  evlsbagval  42554  selvvvval  42573  evlselv  42575  selvadd  42576  prjcrv0  42621